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### Computers in Civil Engineering53:081 Spring 2003

Lecture #6

Roots of Equations(Chapter 5 of 3rd Edition of C2)

Solve for x please …

- 4x=8
- (x-1)(x+1)=0
- e-x=x
- cos(x)cosh(x)+1=0

Problem Formulation

Find roots of equations in form:f(x)=0

(We can put any of our one-equation engineering problems into this form)Question: What do we mean by roots?

Question: Why do we need to find the roots of such equations?

Answer: Example of uniform flow in open channel

Continue...

In many instances Q, So, etc. are known, and one wants to solve for y. In other words: given Q and channel geometry, solve:

f(y) = 0

Problem: Explicit solution does not exist! (especially since A(y) and P(y) are also functions)

Solution:Numerically find root of equation, e.g. find value of depth y such that the equation is satisfied.

(Note that you can move Q to the right side and say 0=f(y). What value of y causes f(y) to be zero?)

Outline

By definition f(xr) = 0

- Single roots
- Multiple roots
- Systems of equations

fi(x1,x2,...,xn)=0 for i = 1,...,n

Graphical Method

- Plot function and observe zero crossing- ALWAYS do this first!
- Useful for finding initial conditions of systematic methods

f(x)

x

General Discussion

- Bracketing Methods
- Require two initial guesses that bracket the true root
- Open Methods
- Require only one initial guess
- Special case: multiple roots. For example:

Two roots at 1

Bracketing Methods

- Bisection Method
- False-Position Method (Regula-Falsi)

Possible Situations

f(x)

f(x)

f(xu)

f(xl)

x

x

f(xl)f(xu) < 0 Odd number of roots in bracket

f(xl)f(xu) > 0 No root in bracket

f(x)

f(x)

x

x

f(xl)f(xu) > 0 Even number of roots in bracket

f(xl)f(xu) < 0 Odd number of roots in bracket

The Bisection Method

1. Choose lower xl and upper xu bounds.

2. Check if f(xl) f(xu) < 0. If not, go to 1.

3. Estimate root by xr = 0.5(xl+xu)

4. Check which interval contains the root.

If f(xl) f(xr) < 0 set xu = xr and go to 5.

If f(xl) f(xr) > 0 set xl = xr and go to 5.

If f(xl) f(xr) = 0 stop. (is this possible?)

5. Is the required accuracy met?

If yes: Stop.

If not: Go to 3.

Termination Criteria

Iteratively estimate error:

In the present context:

Where is the prescribed stopping criterion.

Termination Criteria

Iteration xr|t|% |a|%

1 0.5 11.8

2 0.75 32.2 33.3

3 0.625 10.2 20.0

4 0.5625 0.819 11.1

5 0.59375 4.69 5.3

(The root is approximately at 0.567)

Roots of Equations … to be continued(Chapter 5 of 3rd Edition of C2)

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